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Special Functions
We have come a long way in this module and covered a lot of material dealing with graphs of polynomials.
In this lesson we will look at parent functions and transformations. As you have seen in previous lessons, there is a lot to remember so take good notes and use them as you work through the lesson.
Parent functions are the main function of a function family.
We are very familiar with y = mx + b, but this is not the parent function. The parent function for linear functions is y = x. Some texts say y = mx + b is the parent function with m =1 and b = 0, it is easier to recognize y = x as the parent function and you will see why later.
y = x goes through the origin and has a slope of 1. When we add or subtract a constant (b) we are actually just taking the graph of y = mx and moving it up or down on the coordinate plane, which we call transforming.
We can also change the slope, with or without changing the y-intercept, and this is a transformation also.
Let’s look at some examples.
Parent function: Various Transformations of y = mx + b:
xy
92
1
6
93
2
9
xy
xy
xy
xy
y
xy
Regardless of what we do to the graph – flip it, spin it, move it up or down, it is still a line resembling the parent function.
Parent function: Various Transformations of y = a|x - h| + k:
xy
92
1
6
93
2
9
xy
xy
xy
xy
xy
xy
Parent function: Various Transformations of y = x2:
2xy
959
4
3
2
2
2
2
2
2
xxy
xy
xxy
xy
xy
Parent function: Various Transformations of y = x3:
3xy
6543
955
1
24
3
2
23
3
23
23
3
3
3
xxxy
xxy
xxy
xxy
xxy
xy
xy
Now you should understand that the parent function is the main function and the changes we make to that equation are what transforms the graph.
This is true for all parent functions.
Here are a few more just so you can see the function in it’s simplest form.
**See if you notice anything about the even and odd exponent parent functions
Parent functions:4xy 5xy 6xy
21xy 100xy 301xy
Parent functions:
even isn ,n xy odd isn ,1nx
y
1, bby x 10, bby xeven isn ,1nx
y
odd isn ,n xy
And there are more…but the point is that once we start altering the equations of parent functions we tend to get more and more unique graphs.
The combination of values that can be used make the number of transformations is endless for any one parent function.
How do we keep it all straight?
Rules…of course!
(and many of them!)
k +kCauses the graph to move up d units-kCauses the graph to move down d units
Many of these may seem the same, but the differences are in the placement of the changes (before, inside, or after the parenthesis, absolute value bars, or radicals)!
khbxay )(
aa>1 Causes a vertical stretch by the factor of a0<a<1Causes a vertical compression/ shrink by the factor of a-aA negative in front of a causes a reflection over the x-axis
h+hCauses the graph to move left c units-hCauses the graph to move right c units
bb>1 Causes a horizontal compression by 1/bb<1Causes a horizontal stretch by 1/b -bCauses reflection over the y-axis
Still the same for absolute value functions
aa>1 Causes a vertical stretch by the factor of a0<a<1Causes a vertical compression/ shrink by the factor of a-aA negative in front of a causes a reflection over the x-axis
bb>1 Causes a horizontal compression by 1/bb<1Causes a horizontal stretch by 1/b -bCauses reflection over the y-axis
khbxay k
+kCauses the graph to move up d units-kCauses the graph to move down d units
h+hCauses the graph to move left c units-hCauses the graph to move right c units
khbxay
Still the same for radical functions
aa>1 Causes a vertical stretch by the factor of a0<a<1Causes a vertical compression/ shrink by the factor of a-aA negative in front of a causes a reflection over the x-axis
bb>1 Causes a horizontal compression by 1/bb<1Causes a horizontal stretch by 1/b -bCauses reflection over the y-axis
k +kCauses the graph to move up d units-kCauses the graph to move down d units
h+hCauses the graph to move left c units-hCauses the graph to move right c units
Some clarification:
Any change in a or k (outside the main function) results in change on the y-axis.Any change in b or h (inside the main function) results in a change on the x-axis.
From here – you just have to practice and refer back to this info as you do!
khbxay
khbxay )(
khbxay